To determine which of the graphs shows that the linear system \(3x + 2y = 12\) and \(6x + 4y = 24\) has an infinite number of solutions, we first need to analyze the equations.
Notice that the second equation, \(6x + 4y = 24\), can be simplified by dividing everything by 2:
\[ 3x + 2y = 12 \]
This is exactly the same as the first equation. Therefore, both equations represent the same line. When two equations represent the same line, they share all points along that line, resulting in an infinite number of solutions.
In graphical terms, this situation is represented by two overlapping lines.
From the response options you provided, the graph that illustrates this relationship will show one line completely overlapping another line.
- If you see two lines that are parallel, or completely different (intersecting at one point), those scenarios indicate either no solutions or a unique solution.
- But if the graph shows two lines that lie on top of each other, it confirms that there are infinite solutions.
Thus, you should look for the option that depicts two identical lines (one solid, perhaps, and the other dotted, both fully overlapping) to identify the scenario where the linear system has an infinite number of solutions.